Strong Convergence Theorem for Solving Generalized Mixed Equilibrium Problems and Fixed Point Problems for Total Quasi--Asymptotically Nonexpansive Mappings in Banach Spaces
Zhaoli Ma,1Lin Wang,2and Yunhe Zhao2
Academic Editor: Morteza Rafei
Received09 Feb 2012
Accepted10 Apr 2012
Published12 Jun 2012
Abstract
We introduce an iterative scheme for finding a common element of the set of solutions of generalized mixed equilibrium problems and the set of fixed points for countable families of total quasi--asymptotically nonexpansive mappings in Banach spaces. We prove a strong convergence theorem of the iterative sequence generated by the proposed iterative algorithm in an uniformly smooth and strictly convex Banach space which also enjoys the Kadec-Klee property. The results presented in this paper improve and extend some recent corresponding results.
1. Introduction
Let be a real Banach space with the dual and let be a nonempty closed convex subset of . We denote by and the set of all nonnegative real numbers and the set of all real numbers, respectively. Also, we denote by the normalized duality mapping from to defined by
where denotes the generalized duality pairing. Recall that if is smooth then is single-valued and norm-to-weak* continuous, and that if is uniformly smooth then is uniformly norm-to-norm continuous on bounded subsets of . We will denote by the single-value duality mapping.
A Banach space is said to be strictly convex if for all with . is said to be uniformly convex if, for each , there exists such that for all with . is said to be smooth if the limitexists for all . is said to be uniformly smooth if the above limit exists uniformly in .
Remark 1.1. The following basic properties of Banach space can be founded in [1].(i)If is an uniformly smooth Banach space, then is uniformly continuous on each bounded subset of .(ii)If is a reflexive and strictly convex Banach space, then is norm-weak*-continuous.(iii)If is a smooth, reflexive and strictly convex Banach space, then the normalized duality mapping is single-valued, one-to-one, and surjective.(iv)A Banach space is uniformly smooth if and only if is uniformly convex.(v)Each uniformly convex Banach space has the Kadec-Klee property, that is, for any sequence , if and , then . (See [1, 2]) for more details.
Next, we assume that is a smooth, reflexive, and strictly convex Banach space. Consider the functional defined as in [3, 4] by
It is clear that in a Hilbert space , (1.3) reduces to , for all .
It is obvious from the definition of that
and
Following Alber [3], the generalized projection is defined by
That is, , where is the unique solution to the minimization problem .
The existence and uniqueness of the operator follows from the properties of the functional and strict monotonicity of the mapping (see, e.g., [1β5]). In Hilbert space , .
Let be a nonempty closed convex subset of , let be a mapping from into itself, and let be the set of fixed points of . A point is called an asymptotically fixed point of [6] if there exists a sequence such that and . The set of asymptotical fixed points of will be denoted by . A point is said to be a strong asymptotic fixed point of , if there exists a sequence such that and . The set of strong asymptotical fixed points of will be denoted by .
A mapping is said to be relatively nonexpansive [7β9], if , and , for all , .
A mapping is said to be quasi--nonexpansive, if and , for all , .
A mapping is said to be quasi--asymptotically nonexpansive, if and there exists a real sequence with such that
A mapping is said to be total quasi--asymptotically nonexpansive, if and there exists nonnegative real sequences , with , and a strictly increasing continuous function with such that
A countable family of mappings is said to be uniformly total quasi--asymptotically nonexpansive, if , and there exists nonnegative real sequences , with , and a strictly increasing continuous function with such that for each and each ,
Remark 1.2. From the definition, it is easy to know that:(i)each relatively nonexpansive mapping is closed;(ii)taking , , and then ββ and (1.7) can be rewritten as
this implies that each quasi--asymptotically nonexpansive mapping must be a total quasi--asymptotically nonexpansive mapping, but the converse is not true;(iii)the class of quasi--asymptotically nonexpansive mappings contains properly the class of quasi--nonexpansive mappings as a subclass, but the converse is not true;(iv)the class of quasi--nonexpansive mappings contains properly the class of relatively nonexpansive mappings as a subclass, but the converse may be not true. (See more details [10β14]).
Let be a bifunction, where is the set of real numbers. The equilibrium problem (For short, ) is to find such that
The set of solutions of (1.11) is denoted by .
Let be a nonlinear mapping. The generalized equilibrium problem (for short, ) is to find such that
The set of solutions of (1.12) is denoted by , that is,
Let be a function. The mixed equilibrium problem (for short, ) is to find such that
The set of solutions of (1.14) is denoted by .
The concept generalized mixed equilibrium problem (for short, ) was introduced by Peng and Yao [15] in 2008. is to find such that
The set of solutions of (1.15) is denoted by , that is,
The equilibrium problem is an unifying model for several problems arising in physics, engineering, science optimization, economics, transportation, network and structural analysis, Nash equilibrium problems in noncooperative games, and others. It has been shown that variational inequalities and mathematical programming problems can be viewed as a special realization of the abstract equilibrium problems (e. g., [16, 17]). Many authors have proposed some useful methods to solve the , , , ; see, for instance, [15β23] and the references therein.
In 2005, Matsushita and Takahashi [13] proposed the following hybrid iteration method (it is also called the CQ method) with generalized projection for relatively nonexpansive mapping in a Banach space :
They prove that converges strongly to , where is the generalized projection from onto .
Recently, Qin et al. [24] proposed a shrinking projection method to find a common element of the set of solutions of an equilibrium problem and the set of common fixed points of a finite family of quasi--nonexpansive mappings in the framework of Banach spaces:
where is the generalized projection from onto . They prove that the sequence converges strongly to .
In [25], Saewan and Kumam introduced a modified new hybrid projection method to find a common element of the set of solutions of the generalized mixed equilibrium problems and the set of common fixed points of an infinite family of closed and uniformly quasi--asymptotically nonexpansive mappings in an uniformly smooth and strictly convex Banach spaces with Kadec-Klee property:
where , is the generalized projection of onto . They prove that the sequence converges strongly to .
Very recently, Chang et al. [26] proposed the following iterative algorithm for solving fixed point problems for total quasi--asymptotically nonexpansive mappings:
where , is the generalized projection of onto . They prove that the sequence converges strongly to .
Inspired and motivated by the recent work of Matsushita and Takahashi [13], Qin et al. [24], Saewan and Kumam [25], Chang et al. [26], and so forth, we introduce an iterative scheme for finding a common element of the set of solutions of generalized mixed equilibrium problems and the set of fixed points of a countable families of total quasi--asymptotically nonexpansive mappings in Banach spaces. We prove a strong convergence theorem of the iterative sequence generated by the proposed iterative algorithm in an uniformly smooth and strictly convex Banach space which also enjoys the Kadec-Klee property. The results presented in this paper improve and extend some recent corresponding results in [13, 24β29].
2. Preliminaries
Throughout this paper, let be a real Banach space with the dual and let be a nonempty closed convex subset of . We denote the strong convergence, weak convergence of a sequence to a point by , , respectively, and is the fixed point set of a mapping .
In this paper, for solving generalized mixed equilibrium problems, we assume that bifunction satisfies the following conditions:(A1), for all ;(A2), for all ;(A3) for all , ;(A4) for each , the function is convex and lower semicontinuous.
Lemma 2.1 (see [16]). Let be a nonempty closed convex subset of a smooth, strictly convex, and reflexive Banach space , let be a bifunction from to satisfying (A1)β(A4), and let and , then there exists such that
Lemma 2.2 (see [30]). Let be a nonempty closed convex subset of a smooth, strictly convex, and reflexive Banach space . Let be a continuous and monotone mapping, let be convex and lower semicontinuous and let be a bifunction from to satisfying (A1)β(A4). For and , then there exists such that
Define a mapping as follows:
for all . Then, the following hold:(1) is single-valued;(2) is firmly nonexpansive, that is, ,
(3);(4) is closed and convex;(5), for all and .
Lemma 2.3 (see [28]). Let be a real uniformly smooth and strictly convex Banach space with Kadec-Klee property and let be a nonempty closed convex subset of . Let and be two sequences in such that and , where is the function defined by (1.3), then .
Lemma 2.4 (see [3]). Let be a smooth, strictly convex and reflexive Banach space and let be a nonempty closed convex subset of . Then, the following conclusions hold: (a), for all ;(b)if βand , then if and only if , for all ;(c)for , βif and only if .
Lemma 2.5 (see [28]). Let be a real uniformly smooth and strictly convex Banach space with Kadec-Klee property and let be a nonempty closed convex subset of . Let be a closed and total quasi--asymptotically nonexpansive mapping with nonnegative real sequences , , and a strictly increasing continuous functions such that , (as ) and . If , then the fixed point set of is a closed and convex subset of .
Lemma 2.6 (see [31]). Let be an uniformly convex Banach space, let be a positive number, and let be a closed ball of . Then, for any sequence and for any sequence of positive numbers with , there exists a continuous, strictly increasing, and convex function , such that, for any positive integer , the following holds:
3. Main Results
Theorem 3.1. Let be a nonempty, closed, and convex subset of an uniformly smooth and strictly convex Banach Banach space with Kadec-Klee property. Let be a continuous and monotone mapping and let be a lower semicontinuous and convex function. Let be a bifunction from to satisfying (A1)β(A4). Let be a countable family of closed and uniformly total quasi--asymptotically nonexpansive mappings with nonnegative real sequences , and a strictly increasing continuous function such that , , (as ), and , and for each , is uniformly -Lipschitz continuous. is defined by
where , is the generalized projection of ββonto , ββfor some , ββand ββare sequences in satisfying the following conditions: (1)for each ;(2) for any ;(3) for some . If is a nonempty and bounded subset in , then the sequence ββconverges strongly to , where .
Proof. We will divide the proof into seven steps.Step 1. We first show that and are closed and convex for each . It follows from Lemma 2.5 that is closed and convex subset of for each . Therefore, is closed and convex in . Again by the assumption, is closed and convex. Suppose that is closed and convex for some . Since for any , we know that
Hence, the set is closed and convex. Therefore, and are well defined.Step 2. We show that for all . It is obvious that . Suppose that for some . Since is uniformly smooth, is uniformly convex. By the convexity of , property of , for any given , we observe that
Furthermore, it follows from Lemma 2.6 that, for any positive integers and for any , we have
Substituting (3.4) into (3.3), we get
This shows that . Further, this implies that and hence for all . Since is nonempty, is a nonempty closed convex subset of , and hence exists for all . This implies that the sequence is well defined. Moreover, by the assumption of , , and , from (1.4), we have
Step 3. is bounded and is a convergent sequence. It follows from (3.1) and Lemma 2.4 that
From definition of that and , we have
Therefore, is nondecreasing and bounded. So, is a convergent sequence, without loss of generality, we can assume that . In particular, by (1.4), the sequence is bounded. This implies is also bounded.Step 4. We prove that converges strongly to some point . Since is bounded and is reflexive, there exists a subsequence such that (some point in ). Since is closed and convex and , this implies that is weakly closed and for each . From , we have
Since the norm is weakly lower semicontinuous, we have
and so
This implies that , and so . Since , by virtue of the Kadec-Klee property of , we obtain that
Since is convergent, this together with , we have . If there exists some subsequence such that , then from Lemma 2.4, we have that
This implies that and
Step 5. We prove that . By definition of , we have
Since exists, we have
Since and the definition of , we get
It follows from (3.6) and (3.16) that
From (1.4), we have
So,
This implies that is bounded in . Note that is reflexive and is also reflexive, we can assume that . In view of the reflexive of , we know that . Hence, there exist such that . It follows that
Taking on the both sides of equality above and by the weak lower semicontinuity of norm , we have
That is, , which implies that . It follows that . From (1.4) and the Kadec-Klee property of , we have
Since , so,
Since is uniformly norm-to-norm continuous on bounded subsets of , we obtain
Step 6. We show that . First, we show that . Since , it follows from (3.1) and (3.14) that
Since , by Lemma 2.3,
By (3.3) and (3.4), for any , we have
So,
Therefore,
In view of the property of , we have
Since , this implies that . From Remark 1.1(ii), it yields
Again since
this together with (3.32) and the Kadec-Klee-property of shows that
By the assumption that is uniformly -Lipschitz continuous, we have
This together with (3.34) and shows that and , that is, . In view of the closeness of , it follows that , that is, . By the arbitrariness of , we have . Now, we show that . It follows from (3.2), (3.3), (3.6), Lemma 2.4, and that
By (1.4), we have
Since as , so
Therefore,
Since is reflexive, we may assume that . In view of the reflexive of , we have . Hence, there exist such that . It follows that
Taking on the both sides of equality above yields that
That is, , which implies that . It follows that . Since is norm-weak*-continuous, it follows that . From (3.38) and with the Kadec-Klee property, we obtain
It follows from (3.23) and (3.42) that
Since is uniformly norm-to-norm continuous, we have
By Lemma 2.2, we have
From (A2), we have
Put for all and . Consequently, we get . It follows from (3.46) that
Since is continuous, and from (3.43), and , , as , therefore . Since is monotone, we know that . Further, . So, it follows from (A4), and the weak lower semicontinuity of and (3.43) that
From (A1) and (3.48), we have
and hence
Letting , we have
This implies that . Hence, .Step 7. We prove that . Let . From and , we have
This implies that
By definition of , we have . Therefore, . This completes the proof.
Taking , for each in Theorem 3.1, we have the following result.
Corollary 3.2. Let be a nonempty, closed, and convex subset of an uniformly smooth and strictly convex Banach Banach space with Kadec-Klee property. Let be a continuous and monotone mapping. Let be a bifunction from to satisfying (A1)β(A4). Let be a closed uniformly -Lipschitz continuous and uniformly total quasi--asymptotically nonexpansive mappings with nonnegative real sequences , and a strictly increasing continuous function such that , , (as ), and . Let be the sequence generated by
where , is the generalized projection of βonto , ββand are sequences in , , for some . If ββis a nonempty and bounded subset in , then the sequence converges strongly to , where .
In Theorem 3.1, as , , for each , we can obtain the following corollary.
Corollary 3.3. Let be a nonempty, closed and convex subset of an uniformly smooth and strictly convex Banach Banach space with Kadec-Klee property. Let be a bifunction from to satisfying (A1)β(A4), and be a closed uniformly -Lipschitz continuous and uniformly total quasi--asymptotically nonexpansive mappings with nonnegative real sequences , and a strictly increasing continuous function such that , , (as ) and . Let be the sequence generated by
where , is the generalized projection of βonto , and are sequences in , , ββfor some . If is a nonempty and bounded subset in , then the sequence converges strongly to , where .
Definition 3.4. A countable family of mapping is said to be uniformly quasi--asymptotically nonexpansive, if and there exist real sequences such that for each ,
The following Corollary can be directly obtained from Theorem 3.1.
Corollary 3.5. Let be a nonempty, closed and convex subset of an uniformly smooth and strictly convex Banach Banach space with Kadec-Klee property. Let be a continuous and monotone mapping and let be a lower semicontinuous and convex function. Let be a bifunction from to satisfying (A1)β(A4). Let be an infinite family of closed and uniformly -Lipschitz continuous and uniformly quasi--asymptotically nonexpansive mappings with a sequence , such that is a nonempty and bounded subset in . Let be the sequence generated by
where , is the generalized projection of βonto , for some , and are sequences in . If for all , and for all , then the sequence converges strongly to , where .
Remark 3.6. Theorem 3.1 improves and extend the corresponding results in [13, 24β29] in the following aspects:(1)for the mappings, extend the mappings from relatively nonexpansive mappings, quasi--nonexpansive mappings, and quasi--asymptotically nonexpansive mappings to a countable family of total quasi--asymptotically nonexpansive mappings;(2)for the framework of spaces, extend the space from an uniformly smooth and uniformly convex Banach space to an uniformly smooth and strictly convex Banach space with the Kadec-Klee property.
Acknowledgment
This work was supported by the Natural Science Foundation of Yunnan Province (no. 2011FB074).
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